Let $M$ be a hyperkaehler manifold, and $��$ a closed, positive (1,1)-form which is degenerate everywhere on $M$. We associate to $��$ a family of complex structures on $M$, called a degenerate twistor family, and parametrized by a complex line. When $��$ is a pullback of a Kaehler form under a Lagrangian fibration $L$, all the fibers of degenerate twistor family also admit a Lagrangian fibration, with the fibers isomorphic to that of $L$. Degenerate twistor families can be obtained by taking limits of twistor families, as one of the Kahler forms in the hyperkahler triple goes to $��$.

v. 3.0, 21 pages, referee comments applied, many typos corrected